| L(s) = 1 | + 1.25·2-s − 0.413·4-s + 3.10·5-s − 4.47·7-s − 3.03·8-s + 3.91·10-s + 4.55·11-s − 0.199·13-s − 5.63·14-s − 3.00·16-s − 1.46·17-s + 3.73·19-s − 1.28·20-s + 5.74·22-s + 8.29·23-s + 4.67·25-s − 0.251·26-s + 1.85·28-s + 3.78·29-s + 3.54·31-s + 2.29·32-s − 1.84·34-s − 13.9·35-s − 2.34·37-s + 4.70·38-s − 9.45·40-s + 1.04·41-s + ⋯ |
| L(s) = 1 | + 0.890·2-s − 0.206·4-s + 1.39·5-s − 1.69·7-s − 1.07·8-s + 1.23·10-s + 1.37·11-s − 0.0553·13-s − 1.50·14-s − 0.750·16-s − 0.354·17-s + 0.857·19-s − 0.287·20-s + 1.22·22-s + 1.72·23-s + 0.934·25-s − 0.0492·26-s + 0.349·28-s + 0.702·29-s + 0.636·31-s + 0.406·32-s − 0.316·34-s − 2.35·35-s − 0.385·37-s + 0.763·38-s − 1.49·40-s + 0.162·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.767291941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.767291941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 - 1.25T + 2T^{2} \) |
| 5 | \( 1 - 3.10T + 5T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 - 4.55T + 11T^{2} \) |
| 13 | \( 1 + 0.199T + 13T^{2} \) |
| 17 | \( 1 + 1.46T + 17T^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 23 | \( 1 - 8.29T + 23T^{2} \) |
| 29 | \( 1 - 3.78T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 + 2.34T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 + 0.609T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 + 2.71T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 0.377T + 73T^{2} \) |
| 79 | \( 1 + 0.915T + 79T^{2} \) |
| 83 | \( 1 + 7.42T + 83T^{2} \) |
| 89 | \( 1 + 4.63T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.090550216095462901162545924560, −8.808863245759500262419217961460, −6.90556484773600168976814990539, −6.63588681704298800714554550540, −5.85434856711878491713292484733, −5.22997073622568668344701287587, −4.15410231995515657447027383170, −3.27476995920209265751962860152, −2.59777293753660731757081593013, −0.993802377940641975891393843557,
0.993802377940641975891393843557, 2.59777293753660731757081593013, 3.27476995920209265751962860152, 4.15410231995515657447027383170, 5.22997073622568668344701287587, 5.85434856711878491713292484733, 6.63588681704298800714554550540, 6.90556484773600168976814990539, 8.808863245759500262419217961460, 9.090550216095462901162545924560
